13 ideas
13430 | Infinity: there is an infinity of distinguishable individuals [Ramsey] |
Full Idea: The Axiom of Infinity means that there are an infinity of distinguishable individuals, which is an empirical proposition. | |
From: Frank P. Ramsey (The Foundations of Mathematics [1925], §5) | |
A reaction: The Axiom sounds absurd, as a part of a logical system, but Ramsey ends up defending it. Logical tautologies, which seem to be obviously true, are rendered absurd if they don't refer to any objects, and some of them refer to infinities of objects. |
13428 | Reducibility: to every non-elementary function there is an equivalent elementary function [Ramsey] |
Full Idea: The Axiom of Reducibility asserted that to every non-elementary function there is an equivalent elementary function [note: two functions are equivalent when the same arguments render them both true or both false]. | |
From: Frank P. Ramsey (The Foundations of Mathematics [1925], §2) | |
A reaction: Ramsey in the business of showing that this axiom from Russell and Whitehead is not needed. He says that the axiom seems to be needed for induction and for Dedekind cuts. Since the cuts rest on it, and it is weak, Ramsey says it must go. |
12430 | Classical logic is our preconditions for assessing empirical evidence [Kitcher] |
Full Idea: In my terminology, classical logic (or at least, its most central tenets) consists of propositional preconditions for our assessing empirical evidence in the way we do. | |
From: Philip Kitcher (A Priori Knowledge Revisited [2000], §VII) | |
A reaction: I like an even stronger version of this - that classical logic arises out of our experiences of things, and so we are just assessing empirical evidence in terms of other (generalised) empirical evidence. Logic results from induction. Very unfashionable. |
12431 | I believe classical logic because I was taught it and use it, but it could be undermined [Kitcher] |
Full Idea: I believe the laws of classical logic, in part because I was taught them, and in part because I think I see how those laws are used in assessing evidence. But my belief could easily be undermined by experience. | |
From: Philip Kitcher (A Priori Knowledge Revisited [2000], §VII) | |
A reaction: Quine has one genuine follower! The trouble is his first sentence would fit witch-doctoring just as well. Kitcher went to Cambridge; I hope he doesn't just believe things because he was taught them, or because he 'sees how they are used'! |
13427 | Either 'a = b' vacuously names the same thing, or absurdly names different things [Ramsey] |
Full Idea: In 'a = b' either 'a' and 'b' are names of the same thing, in which case the proposition says nothing, or of different things, in which case it is absurd. In neither case is it an assertion of a fact; it only asserts when a or b are descriptions. | |
From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1) | |
A reaction: This is essentially Frege's problem with Hesperus and Phosphorus. How can identities be informative? So 2+2=4 is extensionally vacuous, but informative because they are different descriptions. |
13334 | Contradictions are either purely logical or mathematical, or they involved thought and language [Ramsey] |
Full Idea: Group A consists of contradictions which would occur in a logical or mathematical system, involving terms such as class or number. Group B contradictions are not purely logical, and contain some reference to thought, language or symbolism. | |
From: Frank P. Ramsey (The Foundations of Mathematics [1925], p.171), quoted by Graham Priest - The Structure of Paradoxes of Self-Reference 1 | |
A reaction: This has become the orthodox division of all paradoxes, but the division is challenged by Priest (Idea 13373). He suggests that we now realise (post-Tarski?) that language is more involved in logic and mathematics than we thought. |
13426 | Formalists neglect content, but the logicists have focused on generalizations, and neglected form [Ramsey] |
Full Idea: The formalists neglected the content altogether and made mathematics meaningless, but the logicians neglected the form and made mathematics consist of any true generalisations; only by taking account of both sides can we obtain an adequate theory. | |
From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1) | |
A reaction: He says mathematics is 'tautological generalizations'. It is a criticism of modern structuralism that it overemphasises form, and fails to pay attention to the meaning of the concepts which stand at the 'nodes' of the structure. |
13425 | Formalism is hopeless, because it focuses on propositions and ignores concepts [Ramsey] |
Full Idea: The hopelessly inadequate formalist theory is, to some extent, the result of considering only the propositions of mathematics and neglecting the analysis of its concepts. | |
From: Frank P. Ramsey (The Foundations of Mathematics [1925], §1) | |
A reaction: You'll have to read Ramsey to see how this thought pans out, but it at least gives a pointer to how to go about addressing the question. |
12428 | Many necessities are inexpressible, and unknowable a priori [Kitcher] |
Full Idea: There are plenty of necessary truths that we are unable to express, let alone know a priori. | |
From: Philip Kitcher (A Priori Knowledge Revisited [2000], §II) | |
A reaction: This certainly seems to put paid to any simplistic idea that the a priori and the necessary are totally coextensive. We might, I suppose, claim that all necessities are a priori for the Archangel Gabriel (or even a very bright cherub). Cf. Idea 12429. |
12429 | Knowing our own existence is a priori, but not necessary [Kitcher] |
Full Idea: What is known a priori may not be necessary, if we know a priori that we ourselves exist and are actual. | |
From: Philip Kitcher (A Priori Knowledge Revisited [2000], §II) | |
A reaction: Compare Idea 12428, which challenges the inverse of this relationship. This one looks equally convincing, and Kripke adds other examples of contingent a priori truths, such as those referring to the metre rule in Paris. |
22328 | I just confront the evidence, and let it act on me [Ramsey] |
Full Idea: I can but put the evidence before me, and let it act on my mind. | |
From: Frank P. Ramsey (The Foundations of Mathematics [1925], p.202), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 70 'Deg' | |
A reaction: Potter calls this observation 'downbeat', but I am an enthusiastic fan. It is roughly my view of both concept formation and of knowledge. You soak up the world, and respond appropriately. The trick is in the selection of evidence to confront. |
22325 | A belief is knowledge if it is true, certain and obtained by a reliable process [Ramsey] |
Full Idea: I have always said that a belief was knowledge if it was 1) true, ii) certain, iii) obtained by a reliable process. | |
From: Frank P. Ramsey (The Foundations of Mathematics [1925], p.258), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 66 'Rel' | |
A reaction: Not sure why it has to be 'certain' as well as 'true'. It seems that 'true' is objective, and 'certain' subjective. I think I know lots of things of which I am not fully certain. Reliabilism long preceded Alvin Goldman. |
15998 | Perfect love is not in spite of imperfections; the imperfections must be loved as well [Kierkegaard] |
Full Idea: To love another in spite of his weaknesses and errors and imperfections is not perfect love. No, to love is to find him lovable in spite of, and together with, his weaknesses and errors and imperfections. | |
From: Søren Kierkegaard (Works of Love [1847], p.158) | |
A reaction: A true romantic at heart, Kierkegaard ideally posits perfect love as unconditional love, and not just of good attributes, predicates and conditions. However, the real question for both me and Kierkegaard is, is perfect love desirable or even possible?[SY] |